# Full text of "Treatise On Analysis Vol-Ii"

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```200       XIII    INTEGRATION

The mapping /i-» j/dA so  defined is clearly a C-linear form on the

complex vector space JS?£(X, |A|) which extends the given C-linear form A
on jf C(X).

(13.16.3)    J/"A /5 dwy complex measure on X, w toe A = /z • |A|, w/zere /z fr a
locally \X\-integrable function such that \h(x)\ = 1  almost everywhere with
respect to |A|.

Let Al9 A2 denote the real measures ^?A, ./A, respectively. Then [AJ <^ |A|
and |A2| ^ m (13.3.7), so that At and A2 are measures with base |A|, and
therefore the same is true of A. If A = h • |A|, it follows that |A| = \h\ - |A|
(13.13.5), which implies that \h(x)\ = 1 almost everywhere with respect
to |A| (13.15.3).

It follows now from the definition (13.16.2) and from (13.14.3) that, for
any A-integrable function/, we have

(13.16.4)                                   f/dA= \fhdW
and hence, by virtue of (13.16.3) and (13.10.3),

(13.16.5)                                /dA

£ (
J

\f\d\X\.

A function is said to be ^-negligible (resp. ^-measurable) if it is |A|-negligible
(resp. | A| -measurable). Likewise for sets.

If A and /z are two complex measures and if /is both A-integrable and
/x-integrable, then / is (A + /i)-integrable, by virtue of (13.16.1) and the
inequality |A + iA S |A| + |/z| (13J.8).

If A is a complex measure, a function g is said to be locally X-integrable
if it is locally |A|-integrable. In that case, for every function /e 3ffc(X), the

function #/is A-integrable, and as in (1 3.1 3) we see that/ h-» | gfdl is a meas-

ure, denoted by g - A. If A = h - |A| (13.16.3), we have jfy dX =jfgh d|A|, or

equivalently g - A = (gh) • |A|, so that the study of measures of the form g - A
is immediately reduced to the case where A is a positive measure. In particular,
it follows from (13.13.4) that

(13.16.6)                            l<rA| = |<7|-|A|.le for all n. Conversely, if/is ^n-integrable for all n, then the func-
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